Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
456 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C89 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C89 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 58. C2×Q8×C89 | Q16 | semidirect |
| 129. #7759 | C89 | direct (N × H) |
| 140. C89 | #7759 | direct (N × H) |
| 193. C2×Q8 | Q16×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2811 | 2 |
| C89×#2024 | 2 |
| C89×#2016 | 2 |
| C89×#1272 | 2 |
| #7759 | 89 |